Wave Optics: YDSE, Diffraction & Polarization

Huygens' Principle

Every point on a wavefront acts as a source of secondary wavelets. The new wavefront is the tangential envelope of these secondary wavelets. This principle explains reflection, refraction, and diffraction qualitatively.

Coherence

Two sources are coherent if they maintain a constant phase difference. Temporal coherence relates to the monochromaticity of the source. Spatial coherence relates to the finite size of the source. Interference requires coherent sources — two independent light sources are never coherent.

Young's Double Slit Experiment (YDSE)

Two slits S₁ and S₂ separated by distance d, screen at distance D:

Path difference: Δx = d sin θ ≈ dy/D (for small θ, where y is the distance from the central maximum on the screen).

Constructive interference (bright fringes): Δx = nλ, where n = 0, ±1, ±2, ...

Destructive interference (dark fringes): Δx = (2n-1)λ/2, where n = 1, 2, 3, ...

Fringe width: β = λD/d (distance between consecutive bright or dark fringes).

Position of nth bright fringe: $y_{n}$ = nλD/d

Position of nth dark fringe: $y_{n}$ = (2n-1)λD/(2d)

Intensity distribution: I = 4I₀ cos²(φ/2), where φ = 2πΔx/λ is the phase difference and I₀ is the intensity from a single slit.

At maxima: $I_{max}$ = 4I₀. At minima: $I_{min}$ = 0 (for equal intensity slits).

For unequal intensities I₁ and I₂: $I_{max}$ = (√I₁ + √I₂)², $I_{min}$ = (√I₁ - √I₂)².

YDSE Modifications

With a glass slab of thickness t and refractive index μ over one slit: Extra optical path = (μ-1)t. Central fringe shifts by Δy = (μ-1)tD/d towards the slab side.

In a medium of refractive index n: Wavelength changes to λ/n, so fringe width β = λD/(nd).

With white light: Central fringe is white. Higher-order fringes show colored edges (violet closer to center, red farther). After a few orders, the pattern disappears due to overlapping.

Diffraction

Single slit diffraction: A slit of width a produces a diffraction pattern with:

• Central maximum: angular width = 2λ/a
• Minima at: a sin θ = nλ (n = ±1, ±2, ...)
• Secondary maxima (approximate) at: a sin θ ≈ (2n+1)λ/2

The central maximum is twice as wide as the secondary maxima. The intensity of secondary maxima decreases rapidly (about 4.5% of central maximum for the first secondary maximum).

Resolving power: Rayleigh criterion — two images are just resolved when the central maximum of one falls on the first minimum of the other. Angular limit: θ = 1.22λ/D.

Polarization

Light is a transverse electromagnetic wave. Polarization restricts the vibration to a single plane.

Malus's Law: When polarized light of intensity I₀ passes through an analyzer at angle θ: I = I₀ cos²θ.

Brewster's Law: When unpolarized light reflects at Brewster's angle $i_{B}$: tan $i_{B}$ = n₂/n₁. The reflected ray is completely polarized (perpendicular to the plane of incidence). The reflected and refracted rays are perpendicular.

Polarization by scattering: Light scattered at 90° to the incident direction is polarized. This is why the sky appears partially polarized.

Types of polarization: Linear (plane), circular, elliptical. A quarter-wave plate converts linear to circular polarization.

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